Definition df-trkg 28431

Description: Define the class of Tarski geometries. A Tarski geometry is a set of points, equipped with a betweenness relation (denoting that a point lies on a line segment between two other points) and a congruence relation (denoting equality of line segment lengths). Here, we are using the following:

• for congruence, (𝑥 − 𝑦) = (𝑧 − 𝑤) where − = (dist‘𝑊)
• for betweenness, 𝑦 ∈ (𝑥𝐼𝑧), where 𝐼 = (Itv‘𝑊)

With this definition, the axiom A2 is actually equivalent to the transitivity of equality, eqtrd 2766.

Tarski originally had more axioms, but later reduced his list to 11:

• A1 A kind of reflexivity for the congruence relation (TarskiG_C)
• A2 Transitivity for the congruence relation (TarskiG_C)
• A3 Identity for the congruence relation (TarskiG_C)
• A4 Axiom of segment construction (TarskiG_CB)
• A5 5-segment axiom (TarskiG_CB)
• A6 Identity for the betweenness relation (TarskiG_B)
• A7 Axiom of Pasch (TarskiG_B)
• A8 Lower dimension axiom (^◡Dim_TarskiG≥ “ {2})
• A9 Upper dimension axiom (V ∖ (^◡Dim_TarskiG≥ “ {3}))
• A10 Euclid's axiom (TarskiG_E)
• A11 Axiom of continuity (TarskiG_B)

Our definition is split into 5 parts:

• congruence axioms TarskiG_C (which metric spaces fulfill)
• betweenness axioms TarskiG_B
• congruence and betweenness axioms TarskiG_CB
• upper and lower dimension axioms Dim_TarskiG≥
• axiom of Euclid / parallel postulate TarskiG_E

So our definition of a Tarskian Geometry includes the 3 axioms for the quaternary congruence relation (A1, A2, A3), the 3 axioms for the ternary betweenness relation (A6, A7, A11), and the 2 axioms of compatibility of the congruence and the betweenness relations (A4,A5).

It does not include Euclid's axiom A10, nor the 2-dimensional axioms A8 (Lower dimension axiom) and A9 (Upper dimension axiom) so the number of dimensions of the geometry it formalizes is not constrained.

Considering A2 as one of the 3 axioms for the quaternary congruence relation is somewhat conventional, because the transitivity of the congruence relation is automatically given by our choice to take the distance as this congruence relation in our definition of Tarski geometries. (Contributed by Thierry Arnoux, 24-Aug-2017.) (Revised by Thierry Arnoux, 27-Apr-2019.)

Assertion

Ref	Expression
df-trkg	⊢ TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))

Distinct variable group:   𝑓,𝑝,𝑖,𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-trkg

Step	Hyp	Ref	Expression
1		cstrkg 28405	. 2 class TarskiG
2		cstrkgc 28406	. . . 4 class TarskiGC
3		cstrkgb 28407	. . . 4 class TarskiGB
4	2, 3	cin 3896	. . 3 class (TarskiGC ∩ TarskiGB)
5		cstrkgcb 28408	. . . 4 class TarskiGCB
6		vf	. . . . . . . . . 10 setvar 𝑓
7	6	cv 1540	. . . . . . . . 9 class 𝑓
8		clng 28412	. . . . . . . . 9 class LineG
9	7, 8	cfv 6481	. . . . . . . 8 class (LineG‘𝑓)
10		vx	. . . . . . . . 9 setvar 𝑥
11		vy	. . . . . . . . 9 setvar 𝑦
12		vp	. . . . . . . . . 10 setvar 𝑝
13	12	cv 1540	. . . . . . . . 9 class 𝑝
14	10	cv 1540	. . . . . . . . . . 11 class 𝑥
15	14	csn 4573	. . . . . . . . . 10 class {𝑥}
16	13, 15	cdif 3894	. . . . . . . . 9 class (𝑝 ∖ {𝑥})
17		vz	. . . . . . . . . . . . 13 setvar 𝑧
18	17	cv 1540	. . . . . . . . . . . 12 class 𝑧
19	11	cv 1540	. . . . . . . . . . . . 13 class 𝑦
20		vi	. . . . . . . . . . . . . 14 setvar 𝑖
21	20	cv 1540	. . . . . . . . . . . . 13 class 𝑖
22	14, 19, 21	co 7346	. . . . . . . . . . . 12 class (𝑥𝑖𝑦)
23	18, 22	wcel 2111	. . . . . . . . . . 11 wff 𝑧 ∈ (𝑥𝑖𝑦)
24	18, 19, 21	co 7346	. . . . . . . . . . . 12 class (𝑧𝑖𝑦)
25	14, 24	wcel 2111	. . . . . . . . . . 11 wff 𝑥 ∈ (𝑧𝑖𝑦)
26	14, 18, 21	co 7346	. . . . . . . . . . . 12 class (𝑥𝑖𝑧)
27	19, 26	wcel 2111	. . . . . . . . . . 11 wff 𝑦 ∈ (𝑥𝑖𝑧)
28	23, 25, 27	w3o 1085	. . . . . . . . . 10 wff (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))
29	28, 17, 13	crab 3395	. . . . . . . . 9 class {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))}
30	10, 11, 13, 16, 29	cmpo 7348	. . . . . . . 8 class (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})
31	9, 30	wceq 1541	. . . . . . 7 wff (LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})
32		citv 28411	. . . . . . . 8 class Itv
33	7, 32	cfv 6481	. . . . . . 7 class (Itv‘𝑓)
34	31, 20, 33	wsbc 3736	. . . . . 6 wff [(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})
35		cbs 17120	. . . . . . 7 class Base
36	7, 35	cfv 6481	. . . . . 6 class (Base‘𝑓)
37	34, 12, 36	wsbc 3736	. . . . 5 wff [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})
38	37, 6	cab 2709	. . . 4 class {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}
39	5, 38	cin 3896	. . 3 class (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})
40	4, 39	cin 3896	. 2 class ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
41	1, 40	wceq 1541	1 wff TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))

This definition is referenced by:  axtgcgrrflx  28440  axtgcgrid  28441  axtgsegcon  28442  axtg5seg  28443  axtgbtwnid  28444  axtgpasch  28445  axtgcont1  28446  tglng  28524  f1otrg  28849  eengtrkg  28964